Denny H. Leung

Some isomorphically polyhedral Orlicz sequence spaces

A Banach space is polyhedral if the unit ball of each of its finite dimensional subspaces is a polyhedron. It is known that a polyhedral Banach space has a separable dual and isc0-saturated, i.e., each closed infinite dimensional subspace contains an isomorph ofc0. In this paper, we show that the Orlicz sequence spacehM is isomorphic to a polyhedral Banach space if limt→0M(Kt)/M(t)=∞ for someK<∞. We also construct an Orlicz sequence spacehM which isc0-saturated, but which is not isomorphic to any polyhedral Banach space. This shows that beingc0-saturated and having a separable dual are not sufficient for a Banach space to be isomorphic to a polyhedral Banach space.

The Bourgain ℓ1-index of mixed Tsirelson space

Abstract Suppose that ( F n ) n=0 ∞ is a sequence of regular families of finite subsets of N such that F 0 contains all singletons, and (θn)n=1∞ is a nonincreasing null sequence in (0,1). The mixed Tsirelson space T( F 0 ,(θ n , F n ) n=1 ∞ ) is the completion of c00 with respect to the implicitly defined norm ||x||= max ||x|| F 0 , sup n∈ N sup θ n ∑ i=1 k ||E i x|| , where ||x|| F 0 = sup F∈ F 0 ||Fx|| l 1 and the last supremum is taken over all sequences (Ei)i=1k in [ N ] such that max E i min E i+1 and { min E i :1⩽i⩽k}∈ F n . In this paper, we compute the Bourgain l1-index of the space T( F 0 ,(θ n , F n ) n=1 ∞ ) . As a consequence, it is shown that if η is a countable ordinal not of the form ωξ for some limit ordinal ξ, then there is a Banach space whose l1-index is ωη.

Some stability properties of c 0 -saturated spaces

A Banach space is

Ideals of operators on (⊕ℓ^{∞}())_{ℓ¹}

c_0

THE NORMED AND BANACH ENVELOPES OF WEAKL 1

-saturated if all of its closed infinite dimensional subspaces contain an isomorph of

The ℓ¹-indices of Tsirelson type spaces

c_0

On order continuous norms

. In this article, we study the stability of this property under the formation of direct sums and tensor products. Some of the results are: (1) a slightly more general version of the fact that

Factoring operators through Hilbert space

c_0

Operators from a subspace of the James space into its dual

-sums of

Fatou Property, representations, and extensions of law-invariant risk measures on general Orlicz spaces

c_0